6.5 Trigonometric Form of a Complex number

De Moivre’s theorem

Graph the Complex number 2+3i

On a Complex Plane

real

imaginary

Absolute Value of the Complex number 4+3i

On a Complex Plane

real

imaginary

53434 22

22

i

babia

Trigonometric Form of a Complex Number

Given: a + bi

r is called the Modulus

θ is the Argument

r

sincos irz

22 bar

abtan

sin,cos rir

Writing a Complex number in Trigonometric form

Point: sin,cos rir

iz 54

Writing a Complex number in Trigonometric form

Point: sin,cos rir

415454

54

22

ir

iz

Writing a Complex number in Trigonometric form

Point: sin,cos rir

7.1283.511803.51

4

5tan

4

5tan

415454

54

1

22

so

ir

iz

3.51

7.128

Writing a Complex number in Trigonometric form

Point: sin,cos rir

7.1283.511803.51

4

5tan

4

5tan

415454

54

1

22

so

ir

iz

7.12841

7.1287.12841

7.128417.12841

cisz

iSinCosz

SinCosz

Product and Quotient of two Complex Numbers

02

2121

2

1

2

1

21212121

2222

1111

z

iSinCosr

r

z

z

iSinCosrrzz

iSinCosrz

iSinCosrz

Find the product

4545cos300

451520

2815

1720

21

2

1

iSin

ciszz

cisz

cisz

Find the product

2150215022

22

300

4545cos300

451520

2815

1720

21

2

1

ii

iSin

ciszz

cisz

cisz

Find the quotient

ii

iSin

ciszz

cisz

cisz

24.5748.294)1908.0(9816.0300

)11()11cos(300

)2817(1520

2815

1720

21

2

1

Power of Complex Numbers

De Moivre’s Theorem

niSinnCosrz

iSinCosrz

iSinCosrz

nn

2222

Power of Complex Numbers

Solve

135216

1354

2

cisz

cisz

Power of Complex Numbers

Solve

iiz

iSinCosz

cisz

cisz

161016

27027016

135216

1354

2

2

2

N th Roots of a Complex Number

Where k = 0, 1, 2, …., n - 1

nk

iSinnk

Cosrn 22

Find the Cube roots of 1000

K = 0, 1, 2..(3-1) done

355

322

322

10

355312

3)1(2

10

10302

302

10

32

32

10003

iiSinCos

iiSinCos

iSinCos

kiSin

kCos

Homework

Page 457 – 459

# 1, 9, 17, 25, 33,

41, 49, 57, 65,

77, 85, 93, 101,

109

Homework

Page 457 – 459 # 5, 13, 21, 29,

37, 45, 53, 61, 73, 81, 89, 97, 105, 113

“I got the Power!!!”